New Family of Centers of Planar Polynomial Differential Systems of Arbitrary Even Degree

Abstract

The problem of distinguish between a focus and a center is one of the classical problems in the qualitative theory of planar differential systems. In this paper we provide a new family of centers of polynomial differential systems of arbitrary even degree. Moreover, we classify the global phase portraits in the Poincaré disc of the centers of this family having degree 2, 4 and 6.

Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

“…which have the circle x 2 + y 2 = r 2 as a periodic orbit surrounding the center (1 − √ 1 + 4r 2 )/2, 0 . The phase portraits in the Poincaré disc of these quadratic polynomial differential systems have been studied in [25], see Figure 1. It is easy to check that system (1) has the nonalgebraic first integral H(x, y) = e −2x (x 2 + y 2 − r 2 ).…”